ES105 - Winter, 2012
Problem Set 7
Due: Friday, March 2th , 11:15 AM. DO ONLY PROBLEMS 1 and 2

Read: Text, Chapter 14

Note: The recommended LAPACK routines for repetitive banded matrix solution via LU factorization are

SGBTRF (computes LU decomposition). Call this to perform the LU factorization only.
SGBTRS (solves assuming SGBTRF has been called). Call this every time you have a new RHS.

Note: All (x,y) units in this problem are stated in the units of the .nod file.

Find the Sources

Given the standard solution to problem 44a (from the last assignment). This is ``TRUTH''. Measurements of voltage are available at the following 12 locations:

along Y=0.20: X=-0.6, -0.3, 0.0, 0.3, 0.6
along Y=0.40: X=-0.3, 0.0, 0.3, 0.6
along Y=0.60: X= 0.0, 0.3, 0.6

These data are available with perfect precision. There is no other data. You want to reconstruct the full solution i.e. the voltage everywhere. Neither the boundary conditions along the bottom "ground plane", nor the location or strength of the sources, are known. The Neumann BC is thought to be perfectly insulated. Prior assumptions:

Along the Dirichlet boundary: the ground condition is known pretty well, within 0.1 volt. The correlation length for the ground condition is relatively short, L=0.4.
Interior sources are presumed to be present but could be anywhere (at any or all nodes). Their expected size is 1.0; they are assumed to have correlation length scale L = 0.3 but they are known to be uncorrelated with the ground.
Model-Data Misfit is expected to be of order .05 volts, with no correlation to anything.

Essentially, you want to find the source distribution and the ground voltage, and use that forcing to construct the voltage everywhere.

Problem 1: TRUTH. Sample the truth to obtain the data. These are perfect data. So a perfect solution exists, if you can find it!
Problem 2: Invert the perfect data using Representers. Make maps of the source distribution, the potential, and the difference between TRUTH and the inverse truth. The latter is an ERROR map -- in this case, you know TRUTH through a special secret arrangement, so the error is computable. Because there is no error in the data, this is a map of BIAS in your least-squares estimator. Compute the RMS of this bias; the RMS of the model-data misfit; the RMS of the nodal sources; and the RMS of the Dirichlet BC's.
Problem 3: Your solution is dependent on the error models used prior to the inversion. Suppose you were unsure about the prior estimate of the model-data misfit. Show the sensitivity of your inverse bias map to this parameter.
Problem 4: Compute the inverse noise assuming observational error will be totally random, uncorrelated, with variance (0.05 volts)**2. Plot contours of the square root of the diagonals of the inverse noise covariance matrix. This is a measure of the IMPRECISION in your estimator.